Month: March 2020

Following on from Show that two points are not connected we have another impossible challenge.

Where the previous article is a critique of existing approaches to knowledge, this challenge goes some way towards showing how things like life, the universe and everything actually work (at least from the perspective of actual life living in an actual universe).

The specific point is that everything is a relationship. It isn’t possible for us to be aware of, or interact with, anything that is not a relationship.

A reasonable starting point is to define what we mean by a relationship.

Except that it is impossible to strictly define anything (dictionaries not withstanding).

You do have some idea of what relationships are. You have come across many relationships in your life. From the relationships between your family and friends to the relationships of physical laws. You know that relationships exist. You are not in anyway surprised that someone would say that there are relationships. The existence of relationships is clear even without a specific definition.

We can make similar observations regarding differences and connections. Clearly there are connections between some things. Obviously some things are different from other things.

What happens when we reverse the question?

Show two things that are not connected.

In the last article I tautologically defined ‘connection’ to be everything you experience.

If you experience something it is (defined to be) connected to everything else you experience.

Using this definition we can speculate that there might exist things that are unconnected to you, but you will never experience those things. Something unconnected to you will never affect you in any way.

Everything that you can experience is connected.

If we assert that things that are connected to each other are related to each other we will find that un-related things also have no bearing on what we experience.

We can observe that things that are connected and related are also different.

So, things that are connected to each other are related to each other. Things that are related to each other are different to each other.

This isn’t very profound yet. Our definitions are deliberate tautologies informed by experience. There are connections, there are relationships and there are observable differences and these things do appear to be closely coupled ideas.

Here comes the twist (I don’t exist)

Well… unlike the urban spaceman, I do exist. However, the things that are connected, related and different – they don’t exist.

More accurately, we only ever see the connection, the relationship and the difference. We cannot see what is being connected, what is related, the object itself.

You exist

Under the current zeitgeist of modern thought you would expect a series of logical arguments based on a set of stated assumptions that you could argue with.

Logic has no place here.

The one thing you are completely certain of is your own existence. As it happens, your own existence encompasses every experience you will ever have. As such, you are absolutely certain of your own experiences.

You experience connections. You know that connections exist. You experience relationships. You know that relationships exist. You experience differences. You know that differences exist.

Things that are not connected to you, that you are not related to, whose difference you cannot perceive – these things are irrelevant to you. They have no significance in your life, they do not form part of your experiences.

The zeitgeist reversed

Conventional wisdom holds that there are things. These things have properties that govern their relationships with other things.

Given that we want to understand and control the relationships between things we need to understand how the properties of the things interact to produce the behaviour we observe.

Gosh that is a lot of redundant effort.

We want to understand the relationships – so why not talk about the relationships directly? Why invent fictional objects that we can never see directly?

The only way to determine the properties of an electron are to observe the relationships over time of an electron.

So… We study the relationships over time of an electron to work out the properties of an electron that will explain the relationships over time of an electron.

No one has ever seen an electron

Never. Ever.

Those electron microscope pictures? They are the result of a whole chain of relationships/interactions.

THERE IS NO SPOON.

Quarks and bosons, the entire particle zoo – all fictions invented to explain the relationships we observe. The relationships exist. Not-relationships do not.

No one has ever seen a particle move. We ‘see’ an interaction and we see another interaction. We then fill in the gaps and assume their must have been a particle and that particle must have moved through a space-time that must exist to justify seeing those two interactions in sequence.

Like frames in a movie, we interpolate distinct events and fill in the gaps with pure imagination.

There are only relationships

(and connections, and differences, and trees made out of relationships, and societies made out of relationships, and planets made out of relationships,…)

The fascinating thing about relationships is that the only way to describe a relationship is in terms of other relationships.

That is all we have ever done. That is all we can ever do. We can manipulate relationships and describe relationships in terms of other relationships.

Since everything we have achieved thus far is ‘just’ playing with relationships it shouldn’t be considered too great a limitation.

You exist

Your existence is completely certain (to you). Your existence is your foundation. Your existence is all the proof you will ever need (or have).

You understand. You find meaning in words. You act and change your world.

There are no words that can prove to you that you don’t exist. There are no words that you can experience that are inconsistent with your experience.

Everything you experience changes you so that you experience things just a little differently in the future.

You are not anybody else. Nobody can understand for you. You have to understand for yourself.

There are only relationships.

I can’t persuade you of this. Nobody can persuade you otherwise.

It is simply part of your existence.

It is self evident that there are only relationships. Granted – this particular bit of self evidence has spent a while wearing a disguise that would make Sherlock Holmes envious.

That disguise is now stripped away.

You can choose to see that you have only ever seen the relationships. Ince you know to look for it it is genuinely as evident and certain as your own existence.

This is, of course, an impossible challenge.

Any two things that you think about or otherwise experience are, at the very least, connected to you (and consequently to each other).

This isn’t a particularly complicated argument: Everything you experience is connected (to some degree) as evidenced by you experiencing them.

We can make this tautologically true by saying that the definition of ‘connection’ is your ability to experience. If you can experience something then it is connected to everything else that you can experience.

Axiomatic mathematics

Axiomatic mathematics (in the broadest sense) is the part of mathematics that provides proofs. One branch of axiomatic mathematics is First Order Logic.

All of axiomatic mathematics is pure mathematics. This means that axiomatic mathematics is not an empirical science. Potentially, a given piece of axiomatic mathematics need have nothing to do with real world observations.

Axiomatic mathematics is abstract in so far as it is not (supposed to be) restricted by the physics of the universe.

Whereas empirical sciences are tested against the universe, axiomatic mathematics needs some other measure of value.

This measure is consistency/inconsistency and is based solely on whether a given argument contains contradictions, or not. (A typical example of a contradiction in mathematics is a statement that turns out to be both true and false at the same time).

Any argument that contains a contradiction can ‘prove’ anything/everything and is consequently as useless as not proving anything (The Principle of Explosion). Such an argument is considered inconsistent.

It is really hard (impossible) to prove that a given argument is consistent.

In practice, a mathematical proof is one where nobody has yet found a contradiction. A proof is typically strengthened if it is based on an existing proof that has already been thoroughly examined for any inconsistency (contradictions).

One such system is ZFC Set Theory (Zermelo–Fraenkel set theory with the axiom of choice).

ZFC Set Theory is an axiomatic system that has been extensively examined by mathematicians and is believed to be consistent (no contradictions have been found after some really smart people looked very hard).

Any statement made using ZFC Set Theory is assumed to be proven automatically by virtue of being made within this ‘known’ consistent axiomatic system.

It is, of course, trivially easy to show that ZFC Set Theory is, in fact, inconsistent, along with every other axiomatic system and all of First Order Logic.

Inconsistency

Any contradiction within an argument (mathematical theory, axiomatic system) renders that whole argument inconsistent.

Any particular contradiction found within an argument is a symptom rather than the disease. One contradiction means that every statement within the argument can be contradicted.

There are no half measures. An argument is either consistent or entirely inconsistent. Any contradiction in any part of an argument renders the entire argument inconsistent.

As such, it is vitally important that two different arguments have no overlap with each other lest a contradiction in one argument renders both arguments inconsistent.

Naturally enough, mathematicians assert that different mathematical arguments (axiomatic systems and logic systems) are unconnected. Each system is judged consistent or inconsistent independent of any other arguments.

This seems fair. Holding a mathematical proof responsible for the faults of a family argument during the holidays seems a bit extreme. Just because one argument is shown to be inconsistent can’t mean that every argument is inconsistent, can it?

Well… yes… if any (mathematical) argument is inconsistent then every mathematical argument is inconsistent.

First, there are many (infinitely many) inconsistent mathematical arguments. It is trivially easy to create known inconsistent mathematical arguments. There is no question that some (mathematical) arguments are inconsistent.

The question is whether one contradiction in one argument applies to all arguments. If all arguments are connected, then any contradiction anywhere means that all arguments are inconsistent.

Given the tautological definition of ‘connection’ given at the top, then clearly all arguments are connected and thus all arguments are inconsistent.

Wait. What?

“There is no way that saying all arguments are connected can demolish huge swathes of modern mathematics (including logic), is there?”

Well actually…

There are a few things to unpack here…

Mathematics is (in part) an attempt to create formal languages that avoid some of the messy miscommunications of natural languages (such as English). As part of that process, mathematics attempts to formalise rules of language and follow those rules far more rigorously than is typical of a normal conversation.

The issue here is not arguments themselves – it is the rules that mathematics thinks should apply to arguments that are at fault.

If you follow the rules of axiomatic mathematics to their logical conclusion, we find that those rules, themselves, are inconsistent by the standards those rules try to impose.

The reason this problem hasn’t previously been glaringly obvious has to do with a confusion between the stated rules of mathematics and the actual mechanics of language.

Language does work – it just doesn’t work in the way that (axiomatic) mathematics would like language to work. As such, much of mathematics works by following the mechanics of natural languages despite claiming to follow a different set of (unworkable) rules. A problem exacerbated by the impression that the rules of mathematics are merely a formalisation of the actual mechanics of language.

Definition of ‘connected’

In natural language we have no problem considering two separate arguments to be unconnected.

Your family squabble over who last saw the TV remote is clearly different to a discussion about how to get the best thrust to weight ratio for a rocket engine.

Even knowing that both those arguments are connected by being between humans living on planet Earth doesn’t change your perception that they are two different arguments.

The rules of mathematics, however, are rather more stringent. Specifically, if something is true and false at the same time it is inconsistent – useless.

In this case we know that all arguments are connected (by the definition of connected given above). We also know that mathematicians have declared particular arguments to be unconnected.

If this holds then within mathematics, arguments are both connected and unconnected. This is a contradiction and renders anything and everything that follows from that contradiction inconsistent.

Formal languages are, by intent, quite technical in so much as some words/meanings have a special and/or narrow meaning that doesn’t necessarily translate well to more natural uses of those words.

Perhaps mathematics has a more rigorous/technical definition of ‘connection’.

It is certainly the case that I chose a (tautological) definition of ‘connected’ that most clearly makes the point.

Dirty tricks – or how to be too clever by half

Definitions are tricky. Clearly one of the intents of mathematics is to clearly define terms, but this turns out to be (again, impossibly) hard.

There is, however a workaround.

Mathematicians don’t need to define what they mean by ‘connected’ or ‘unconnected’. Instead they can declare that there is a definition of ‘connected’ that is consistent with their requirements.

They can then use that abstract definition and, so long as that abstract definition doesn’t lead to a contradiction, then they can assume that their declaration is consistent and consequently legitimate.

There are many, many contradictions that follow from the rules of axiomatic mathematics. But since we can declare each of those contradictions to be unconnected to anything else (including the rules of mathematics), those contradictions don’t count.

So long as there might be a definition of ‘connected’ that serves to isolate a given contradiction from a given argument then no contradiction disproves axiomatic mathematics.

Bonus points, because ‘connected’ hasn’t actually been defined, any definition that shows axiomatic mathematics to be inconsistent is the wrong definition of ‘connected’ and can be disregarded.

Even better, ‘connected’ can mean whatever anyone needs at any time because there is no way to show that it doesn’t mean whatever you want it to mean. There is no way for anyone to prove that you are using the word incorrectly or inconsistently.

In short, it is completely impossible to challenge the foundations of axiomatic mathematics using the standard tools of logic (a branch of axiomatic mathematics).

To be fair

Asserting that there is a definition that satisfies the requirements is quite clever.

It really is remarkably hard (impossible) to nail a single, unambiguous definition to a word. At the same time it seems perfectly obvious that we do define words. There just appears to be an annoying gap between fairly fuzzy natural language definitions and the really rigorous and precise definitions that would be so useful for formal languages.

Assuming the existence of the desired definition and then behaving as if we had that definition seemed to be working remarkably well.

Really, seriously. Mathematicians know that the approach isn’t perfect (The Foundational Crises in Mathematics). However, quite a lot of mathematics seems to be working really well; and whatever slight drawbacks it may have, what is the alternative? Nobody is throwing away something (however flawed) in favour of nothing.

Mathematics had to do some creative accounting just to get started (the process was somewhat more ad hoc with quite a lot of this being justification after the fact but…).

On the other hand

Mathematics finds itself locked into a set of mistaken assumptions that are immune to the very tools of mathematics that are supposed to expose mistaken assumptions

The standard tool of reason (logic) is incapable of a critical level of self analysis.

It is a bit of a hint in itself that axiomatic mathematics has to be made immune to the rules of axiomatic mathematics before it even gets started.

In any case, there cannot be proof that the assumptions of axiomatic mathematics are false because those assumptions have been isolated from the mechanisms of proof.

You can, however, experience for yourself the evidence that everything you experience is connected.

Try the headline challenge for yourself. Feel free to change the definition of ‘connected’.

You will find that it is obvious/clear that by any reasonable definition, all of your experiences are connected to each other to some degree.

It is quite impossible to create a definition of connection that you can

1. Communicate to other people.
2. That clearly shows some statements are connected within an argument.
3. Shows that any given pair of arguments are not connected.